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Theorem ruv 8663
 Description: The Russell class is equal to the universe V. Exercise 5 of [TakeutiZaring] p. 22. (Contributed by Alan Sare, 4-Oct-2008.)
Assertion
Ref Expression
ruv {𝑥𝑥𝑥} = V

Proof of Theorem ruv
StepHypRef Expression
1 df-v 3353 . 2 V = {𝑥𝑥 = 𝑥}
2 equid 2097 . . . 4 𝑥 = 𝑥
3 elirrv 8657 . . . . 5 ¬ 𝑥𝑥
43nelir 3049 . . . 4 𝑥𝑥
52, 42th 254 . . 3 (𝑥 = 𝑥𝑥𝑥)
65abbii 2888 . 2 {𝑥𝑥 = 𝑥} = {𝑥𝑥𝑥}
71, 6eqtr2i 2794 1 {𝑥𝑥𝑥} = V
 Colors of variables: wff setvar class Syntax hints:   = wceq 1631  {cab 2757   ∉ wnel 3046  Vcvv 3351 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-reg 8653 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-nel 3047  df-ral 3066  df-rex 3067  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4317  df-pr 4319 This theorem is referenced by:  ruALT  8664
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